A poll of 1,255 people for campaign group Ecpat UK and the Body Shop found one in 10 did not know that children are trafficked into the UK.
The survey also found that 22% admitted buying fake DVDs or visiting brothels, which can perpetuate child trafficking.
The Child Exploitation and Online Protection Centre identified some 325 potential victims in 2007 to 2008.
The Home Office said the hidden nature of the trade in children made it extremely difficult to estimate the scale of the problem.
'Hard to spot'
The study showed 34% of people believe trafficked children end up in foreign countries, not in Britain.
However, a government Child Exploitation and Online Protection Centre (Ceop) report in April found that victims had come to the UK from 52 different countries...
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